Mesonic correlation functions at finite temperature and density in the Nambu–Jona-Lasinio model with a Polyakov loop

We investigate the properties of scalar and pseudoscalar mesons at finite temperature and quark chemical potential in the framework of the Nambu–Jona-Lasinio (NJL) model coupled to the Polyakov loop (PNJL model) with the aim of taking into account features of both chiral symmetry breaking and deconfinement. The mesonic correlators are obtained by solving the Schwinger-Dyson equation in the RPA approximation with the Hartree (mean field) quark propagator at finite temperature and density. In the phase of broken chiral symmetry, a narrower width for the $\sigma$ meson is obtained with respect to the NJL case; on the other hand, the pion still behaves as a Goldstone boson. When chiral symmetry is restored, the pion and $\sigma$ spectral functions tend to merge. The Mott temperature for the pion is also computed.

Figure 1

Effective potential in the pure-gauge sector [Eq. (6)] for two characteristic temperatures, below and above the critical temperature $T_0$. One can see three minima appearing above $T_0$.

Figure 2

PNJL (solid line), NJL (dotted line), and lattice results (points) for the net quark density at $\mu =0.6T_c$ (from 35).

Figure 3

Fermi-Dirac distribution function $f(E_p-\mu )$ (valid for the NJL model) and the corresponding function $f_{\Phi }^{+}(E_p)$ (valid for the PNJL one) as functions of $p$, for different temperatures. $\Phi$, $\overline{\Phi }$ and $m$ are taken at their mean field values. The upper lines refer to $T=0.3\mathrm{GeV}$, and the lower ones to $T=0.1\mathrm{GeV}$.

Figure 4

Top panel: Masses of the $\sigma$ and $\pi$ as functions of the temperature, together with the Hartree quark mass and the Polyakov loop, in the PNJL model at $\mu =0$. The threshold $2m$ is also plotted to show that the $\sigma$ mass is close to this value below 0.25 GeV. Bottom panel: Same as before, but without the Polyakov loop and adding instead the width of the mesons, represented by error bars.

Figure 5

Masses of the $\sigma$ and $\pi$ as functions of the temperature, together with the Hartree quark mass, in the PNJL model at $\mu =0.27\mathrm{GeV}$ (top panel) and $\mu =0.34\mathrm{GeV}$ (bottom panel). The lower figure clearly displays a first order phase transition related to the discontinuity of the chiral condensate occurring at $T_c^{\chi }\simeq 0.06\mathrm{GeV}$.

Figure 6

Spectral function $F^{MM}(\omega )$ of the pion (top panel) and sigma (bottom panel) in PNJL, at $\overrightarrow{q}=\overrightarrow{0}$, as a function of $\omega$, for different temperatures and $\mu =0$. The width of the delta peak in the $\pi$ spectral function when $m_{\pi }<2m$ is artificially given by a small, positive $i\eta$.

Figure 7

$\sigma$ and $\pi$ masses as functions of the temperature, together with the Hartree quark mass in the NJL (thin lines) and PNJL (thick lines) models ($\mu =0$).

Figure 8

Comparison of the $\sigma$ spectral function at $\overrightarrow{q}=\overrightarrow{0}$, as a function of $\omega$, for different temperatures and two chemical potentials. Thick lines correspond to the PNJL model and thin ones to the NJL model. Notice the almost perfect coincidence at $T=T_c^{\sigma \mathrm{\text{−}}\min }$.

Figure 9

Comparison of the width of the $\sigma$ in NJL and PNJL as a function of the reduced temperature $T/T_c$. Top panel: absolute comparison at zero chemical potential. Bottom panel: relative width ${\Gamma }_{\sigma }(\mathrm{PNJL})/{\Gamma }_{\sigma }(\mathrm{NJL})$ at $\mu =0$ and $\mu =0.27\mathrm{GeV}$. In the latter, a maximum effect below $T_c$ ($T\simeq 0.87T_c$) of $\simeq 40\%$ can be seen.